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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>2. For</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Eq3_1.html ./knowl/Eq3_1.html ./knowl/Eq3_1.html">
\begin{equation*}
Y(s)=\frac{s^2}{s^4-1},
\end{equation*}
</div>
<p class="continuation">a partial fraction expansion of <span class="process-math">\(Y(s)\)</span> is</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Eq3_1.html ./knowl/Eq3_1.html ./knowl/Eq3_1.html">
\begin{equation*}
Y(s)=\frac{as+b}{s^2-1}+\frac{cs+d}{s^2+1}.
\end{equation*}
</div>
<p class="continuation">It follows that</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Eq3_1.html ./knowl/Eq3_1.html ./knowl/Eq3_1.html">
\begin{equation}
(as+b)(s^2+1)+(cs+d)(s^2-1)=s^2\tag{8.3.3}
\end{equation}
</div>
<p class="continuation">for all <span class="process-math">\(s\text{.}\)</span> By setting <span class="process-math">\(s=1\)</span> and <span class="process-math">\(s=-1\)</span> respectively in (<a href="" class="xref" data-knowl="./knowl/Eq3_1.html" title="Equation 8.3.3">(8.3.3)</a>), we have</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Eq3_1.html ./knowl/Eq3_1.html ./knowl/Eq3_1.html">
\begin{equation*}
2(a+b)=1,\quad 2(-a+b)=1,
\end{equation*}
</div>
<p class="continuation">and therefore <span class="process-math">\(a=0, b=1/2\text{.}\)</span> If we set <span class="process-math">\(s=0\)</span> in (<a href="" class="xref" data-knowl="./knowl/Eq3_1.html" title="Equation 8.3.3">(8.3.3)</a>), then <span class="process-math">\(b-d=0\text{,}\)</span> so <span class="process-math">\(d=1/2\text{.}\)</span> Finally, equating the coefficients of the cubic terms on each side of (<a href="" class="xref" data-knowl="./knowl/Eq3_1.html" title="Equation 8.3.3">(8.3.3)</a>), we find that <span class="process-math">\(a+c=0\text{,}\)</span> so <span class="process-math">\(c=0\text{.}\)</span> Thus</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Eq3_1.html ./knowl/Eq3_1.html ./knowl/Eq3_1.html">
\begin{equation*}
Y(s)=\frac{1/2}{s^2-1}+\frac{1/2}{s^2+1}
\end{equation*}
</div>
<p class="continuation">and</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Eq3_1.html ./knowl/Eq3_1.html ./knowl/Eq3_1.html">
\begin{equation*}
y(t)=\frac{\sinh t+\sin t}{2}.
\end{equation*}
</div>
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